Proof theory is an area of logic that studies proof as formal mathematical objects. If you'd like advice on the presentation of a proof you have in draft, use proof-writing instead. If you'd like feedback on its validity, use proof-verification. If none of the above apply, you do not need a proof-* ...

learn more… | top users | synonyms

1
vote
0answers
33 views

Is the full strength of first-order logic needed for dealing with equational theories?

More specifically, if we have an equational theory $T$ (a set of equations understood as being implicitly universally quantified), are the (equational) consequences of $T$ that can be proved with ...
3
votes
1answer
77 views

What is the least ordinal $\beta$ for which the function $f_\beta(n)$ in fast-growing hierarchy is incomputable?

Fast-growing hierarchy consists of a transfinite succession of faster growing functions $f_\alpha$: $f_0(n) := n+1$, $f_{\alpha+1}(n) := f^n_\alpha(n)$, $f_{\alpha}(n) := f_{\alpha[n]}(n)$ if ...
0
votes
1answer
65 views

More details of the “Standard View og Proof” with three points are needed.

I have a Danish book about the theory of knowledge for mathematicians which I have tried my best to translate some parts into English. According to the lecturer, we can with "certain reasonability" ...
22
votes
3answers
2k views

Why is Gödel's Second Incompleteness Theorem important?

Given that the consistency of a system can be proven outside of the given formal system, Gödel says, It must be noted that proposition XI... represents no contradiction to the formalities ...
6
votes
6answers
172 views

Notable examples of “impossible” results ruled out by earlier barrier or no-go theorems or widespread beliefs

there is a certain style of type of proof in mathematics something like a "barrier theorem" but which also relates to widespread mathematical beliefs/ "conventional wisdom". an example would be the ...
0
votes
1answer
47 views

Inversion lemma for G3ip

I'm following the book Structural Proof Theory by Negri and others. In it, they claim on page 32 about G3ip that if $⊢ _ n A \& B, Γ ⇒ C$, then $⊢ _ n A, B, Γ ⇒ C$. But, given that the only ...
2
votes
1answer
139 views

What are 2-morphisms in the category of proofs?

After reading through "Categories for the practicing physicist" I came to learn there is a category whose objects are propositions $A,B,...$ and whose morphisms are proofs $f:A\rightarrow{B}$ that ...
1
vote
1answer
64 views

Is the length of the proof of propositional tautology a PA-total function?

Suppose we have fix some interpretation of propositional (not first-order!) logic inside PA, and say $f(n) = $ {the maximum length of a proof of a tautology with $n$ propositional primitives} ...
2
votes
1answer
86 views

Couple of questions from Takeuti's Proof Theory book

I am reading Gaisi Takeuti's Proof Theory (Second Edition, Dover), and I have a couple of questions: I) Right after the first (1.1.) definition, the author says that "In any case it is essential that ...
0
votes
1answer
33 views

Left “ and” inference rule G3cp

If you are at the stage of $$((P\supset Q) \wedge (Q\supset \bot)) \Longrightarrow (P \supset \bot)$$ And then you apply an $L\wedge$ rule, Why do you get $$(P \supset Q),(Q\supset\bot)),P ...
0
votes
1answer
31 views

Example of an axioms in G3Cp

Would $$A,B,C \Longrightarrow C,A$$ be an axiom in G3Cp? I ask because B is not on both sides and I am not sure if that makes a difference or not?
1
vote
1answer
48 views

What is the proof-theoretic ordinal of $PA+TI(\epsilon_0)$?

what is the proof-theoretic ordinal for $PA+TI(\epsilon_0)$, where $PA+TI(\epsilon_0)$ is Peano arithmetic where transfinite induction up to $\epsilon_0$ was added? Is it known? Thank you
1
vote
0answers
24 views

Independence of FLT over weak systems

It is known that Fermat's last theorem can be proven in finite-order arithmetic (e.g. accoridng to this site). This is still an extremely high upper bound on proof complexity (for example, compared to ...
0
votes
0answers
76 views

Are proofs for many-sorted first order logic shorter than single sorted first order logic?

I understand that the expressive power of first order logic with one sort is the same as any many sorted first order logic, and that higher order logic with general semantics is the same as a many ...
3
votes
2answers
114 views

What is the fastest growing primitive-recursive-function?

Fast growing functions tend to be not primitive-recursive. So I wonder if there is a limit how fast a function can grow, if it is known that it is primitive recursive. What is the fastest growing ...
0
votes
1answer
114 views

What is the slowest growing function that cannot be proven to be total by PA?

I asked the question if PA can prove any function growing faster than $f_{\epsilon_0}(n)$ to be total. The answer was no. What about the converse : Can prove PA every function growing slower than ...
0
votes
2answers
57 views

Inference rule for Non-Empty Domains

I am currently experimenting with logic frameworks. I am basically using something along dependent types as in "Proof-assistants using Dependent Type Systems" by Henk Barendregt and Herman Geuvers. ...
4
votes
2answers
87 views

Is there any unreachable result?

I hope that this question is reasonable and make sense because I am not sure. Every theorem's proof is consisting of finite logical steps. Can a proof of the theorem require infinitely many ...
3
votes
0answers
79 views

Is the following derivation of Gödel's Paradox generalizable to an intuitionistic one using provability?

I will follow Goedel's original proof of Goedel's paradox located on pg. 264 of Hao Wang's A Logical Journey: From Gödel to Philosophy. With this in sight, and to also avoid confusion, I will also ...
7
votes
1answer
69 views

Can PA prove very fast growing functions to be total?

The Goodstein-sequence is a total function, but PA cannot prove this. Is this true for any other function with growth rate at least $f_{\epsilon_0}$ or are there functions growing at least as fast ...
1
vote
1answer
83 views

Proof in sequent calculus without cut

I met an exercise in Gaisi Takeuti, Proof Theory [Exercise 2.7, page 14]. How to construct a cut-free proof of$\ \forall xA(x)\rightarrow B\vdash \exists x(A(x)\rightarrow B)$, where A(a) and B are ...
1
vote
1answer
152 views

2 Questions regarding Relative Consistency Proofs

First Question: Let IC be the statement "There is an inaccessible cardinal." I have read that one cannot prove (in ZFC) the relative consistency of ZFC + IC w.r.t. ZFC. i.e. $ Con(ZFC) \rightarrow ...
1
vote
2answers
32 views

What does this proposition mean?

$∀x ∈ P(\Bbb{N}), x \notin \{\} \Rightarrow ∃y ∈ x, ∀z ∈ x \ | \ y < z$ Where $P(x)$ is the power set. I'm interpreting it as "in all subsets of the natural numbers, there exists a value smaller ...
1
vote
1answer
173 views

How to check homeomorphic embedding relation programmatically?

This is a follow up to this question and Deedlit's answer. I'm looking for a precise definition of the "hem?" (tree A homeomorphically embeddable in tree B?) relation, preferably in terms of a ...
-2
votes
1answer
50 views

Z / 6Z being a set of well dedfined equivalence classes, and a congruent to b(mod 6)

why is this = [0],[1],[2],[3],[4],[5],[6] and how would I define f Z/6Z - Z/6Z by f([a]) = ([2a]). I have the proof but I don't understand it. Proof: Assume [a1] = [a2] in Z/6Z. then a1 congruent to ...
0
votes
1answer
76 views

Proving Theorem: subspace of polynomials of degree two or less?

How can I prove that the set $S$ of polynomials of degree $2$ or less, whose coefficients sum to zero, is a subspace of all polynomials with degree $2$ or less? I know I need to show that $a+b+c=0$ ...
1
vote
5answers
116 views

Proof by Induction Divisibility.

$6^n-5n+4$ is divisible by 5 for all positive integers $n$. $n >=1$ Prove By Induction My attempt is as follows: $n=1$ $6^1-5(1) +4$ $=5$, Therefore 5 is divisible by 5 so $n=1$ is true ...
1
vote
1answer
100 views

Uncountable reals in the theory

The Question I'm looking for a possibility to somehow proof the "essence" of Cantor's diagonal argument within a recursive first-order theory which is satisfied by the reals (better: within a theory ...
2
votes
0answers
63 views

How far can-I rewrite in lambda functions?

I am quite new with the lambda calculus. I am experimenting lambda-calculus proofs through the coq proof assistant, but the question I have is not related to coq (I guess). However, I'm going to use ...
0
votes
1answer
95 views

Euclid's proof for infinitely many prime numbers

Prove that there are infinitely many primes congruent to 3mod4 using euclid's proof for infinitely many prime number. I guess I don't really know where to start because I don't understand euclid's ...
1
vote
1answer
76 views

Comparing Statements and predicates using Truth Tables

Consider the four statements: $∃x$ $∀y$ $p(x, y)$ $∃y$ $∀x$ $p(x, y)$ $∀x$ $∃y$ $p(x, y)$ $∀y$ $∃x$ $p(x, y)$ which we call S1, S2, S3 and S4 respectively. Does there exist a predicate p such ...
1
vote
1answer
60 views

Model-theoretic question about language of field theory.

Let $\mathscr{L}=\{+,·\}$ be the language of the theory of fields. Let $\phi$ be a sentence in this language. Show, using the compactness theorem of first-order logic, that if $\phi$ holds in finite ...
13
votes
2answers
1k views

How to find the shortest proof of a provable theorem?

Roughly speaking, there are some fundamental theorems in mathematics which have several proofs (e.g. Fundamental Theorem of Algebra), some short and some long. It is always an interesting question ...
3
votes
2answers
74 views

Undecidability and truth

Are there undecidable problems for which a single truth exists? For example, the question about parallels is not decidable from Euclid axioms. But multiple answers are valid and give different kinds ...
0
votes
2answers
65 views

Prove the Cauchy-Schwarz Inequality (missing a step)

during lecture notes I only caught most of the proof and couldnt write a step down fast enough, and I'm having a touch trouble seeing how to get from the previous step to the next. Here is what i have ...
1
vote
1answer
76 views

How can Goodstein's theorem be expressed in PA

I understand Goodstein's Theorem and its proof. I'm trying to understand the proof of why Goodstein's Theorem cannot be proved in PA. However, it's not immediately clear to me that Goodstein's Theorem ...
8
votes
1answer
483 views

Why wouldn't someone accept Gentzen's consistency proof?

Reading the consistency section of the Peano Axioms wikipedia page, I came across this sentence: The vast majority of contemporary mathematicians believe that Peano's axioms are consistent, ...
1
vote
1answer
68 views

Question about the incompleteness proof (Theorem V)

Question in short: Where do I find a complete proof of Theorem V from Gödels incompleteness proof? If it does not exists, can someone provide it? Question in detail: I am trying to understand ...
4
votes
1answer
196 views

Birkhoff's completeness theorem

I have two simple questions. A) Does Birkhoff's completeness theorem follow directly from Gödel's completeness theorem? B) Is Birkhoff's completeness theorem constructive in the following sense: ...
3
votes
2answers
212 views

Is constructive proof of non-existence possible

Constructive proof construct(indicates) object that satisfies given predicate. Question is whether one can give constructive proof of non-existence of an object with given property e.g. that every ...
2
votes
2answers
427 views

Weakening and Contraction

I saw this site saying weakening is a structural rule where the hypotheses or conclusion of a sequent may be extended with additional members and that contraction is a rule where two equal (or ...
0
votes
1answer
118 views

What is really a “complete” deductive system for first-order theories.

Given some first-order language and a set of axioms therefrom one still needs to specify a deductive system to turn it into a full-fledged first-order theory. Currently I'm under the following ...
1
vote
0answers
64 views

Is a sentence in $\Pi_1$ true given $Q \vdash \lnot\varphi$?

If $Q \vdash \lnot\varphi$ (Q is the Robinson arithmetic), and if I assume that $\varphi \in \Pi_1$; Can I say that $\varphi$ is a true sentence? My thoughts are that, given that Q is $\Sigma_1$- ...
1
vote
1answer
78 views

A simple question on Gödel's functional interpretation

I've been recently reading the Gödel's functional interpretation (or Dialectica). It is generally defined inductively, as could be found here: http://www.andrew.cmu.edu/user/avigad/Papers/dialect.pdf ...
3
votes
1answer
304 views

Problems with nesting proof predicates in first order logic.

Whenever I start nesting proof predicates, I always seems to run into these bizarre situations. I was wondering if anyone knows about this and could shed some light on it or provide me with some ...
5
votes
2answers
219 views

Is there a minimal axiomatization of ZFC?

Working in ZFC, does there exist a set $\Sigma$ of sentences which axiomatizes ZFC (i.e. every sentence in $\Sigma$ is provable from your favorite axiomatization of ZFC, and vice versa) and is minimal ...
11
votes
3answers
203 views

The category of theorems and proofs

On a philosophy website, it said that you could have a category with theorems as objects and proofs as arrows. This sounds awesome, but I couldn't find anything on the web that has both "category" and ...
4
votes
2answers
268 views

What if a conjecture were provably unprovable?

Suppose we found a proof that "The Twin Prime Conjecture cannot be proven", without any conclusion as to the conjecture itself being true or false. Is it then possible for the conjecture to be true? ...
9
votes
1answer
146 views

Is there a way to tell how many different ways you can prove a theorem?

Consider the question. Given the nature of a sentence $S$, it there any way to tell how many different ways you can prove this sentence? Proofs are not distinct if we have a situation such as: $P ...
1
vote
2answers
127 views

Is it possible to prove that the encoding of existentials in System F is valid?

In Girard's Proofs and Types, under item 11.3.5, second-order existential quantification is encoded in System F using universal quantification as follows: $$ \Sigma X.V \equiv \Pi Y. (\Pi X.(V \to ...